The friendship paradox is the idea that your friends have more friends than you do, which turns out to be true for most people. It may seem counterintuitive, but there is plenty of evidence to back up the claim and a simple mathematical analysis shows why it is true.

The fact that people’s friends are more popular than they are may also explain another observation for which there is growing evidence—that excessive use of social networks makes people less happy. It’s easy to imagine that knowing that they are less popular than their friends makes people less happy.

This has led to widespread speculation that the distribution of happiness throughout a social network might also lead to a happiness paradox. If happiness correlates with popularity—the being popular makes people happy—then this could be true too.

That’s an interesting hypothesis but there has never been strong evidence to back it up. Until now.

Today that changes thanks to the work of Johan Bollen at Indiana University in Bloomington and a few pals, who have found the first evidence of a happiness paradox on Twitter. They say that it is good evidence that social network use can affect the well-being of a significant proportion of the planet’s population.

The friendship paradox is straightforward to explain. It comes about because of the skewed way people collect friends on online social networks such as Twitter and Facebook. Most people have a small number of friends—a few dozen or so. But a tiny fraction of people have huge numbers of friends millions or tens of millions of followers in some cases.

This has two effects. First, it makes them much more likely to appear in a random person’s list of friends. And second, it dramatically skews the answer when calculating the average number of friends that a person’s friends have.

To see how, imagine you have 10 friends and that nine of them also all have 10 friends each. But one of your friends has a million friends. In that case, your friends have close to 100,000 friends each on average while you have only 10. And all your friends are in the same position, except the one who has a million friends.

If popularity correlates with happiness, then it’s reasonable to expect that a happiness paradox might also be observable. And that’s exactly what Bollen and co find but with an unexpected twist.

Bollen and co begin by analyzing the most recent 3,000 tweets sent by some 40,000 Twitter users. They use a standard algorithm to analyze each tweet to determine its sentiment—whether positive or negative—and then assume this gives a sense of the user’s happiness level. In other words, they assume that people who are less happy send more negative tweets. They also include in the analysis the number of followers and followees for each individual.

The results make for interesting reading. Bollen and co say there is clear friendship paradox at work in this network, as expected. But they also say there is a less striking but nonetheless significant happiness paradox at work, too.

Indeed, Bollen co say their evidence suggests that the more unhappy the individual, the stronger the happiness paradox they face. “Although happy and unhappy groups of subjects are both affected by a significant happiness paradox, unhappy subjects are most strongly affected,” they say.

That’s something of a surprise. Unhappy people also seem to experience a less significant friendship paradox, so it’s easy to think the same would be true of the unhappiness paradox. But not so.

Bollen and co think they know why. “A possible explanation may lie in the stronger relation between the happiness of individuals in this group and the overall happiness of their friends,” they say.

That’s interesting because it suggests a different origin for the paradox. “Instead of resulting from the greater prevalence of popular and happy individuals, it could come about by the social interactions between people. In other words, unhappiness is more infectious than happiness for certain individuals.

This work has some important limitations. The most obvious is the possibility that the sentiment of a tweet is not an accurate reflection of the sender’s happiness levels. Perhaps people who send negative tweets are happier because they get negative thoughts off their chests?

Nevertheless, this is interesting work that highlights the profound influence that social networks have over our well-being. We know that tweets spread around the network, sometimes explosively over huge distances. Could it be possible that happiness also sweeps the planet in the same way, carried on a tide of positive sentiment in tweets? And if so, could tidal waves of unhappiness flood the planet influencing billions of people in the process?